\(\int x^2 (a^2+2 a b x^2+b^2 x^4)^3 \, dx\) [450]

Optimal result

Integrand size = 24, antiderivative size = 82 \[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {a^6 x^3}{3}+\frac {6}{5} a^5 b x^5+\frac {15}{7} a^4 b^2 x^7+\frac {20}{9} a^3 b^3 x^9+\frac {15}{11} a^2 b^4 x^{11}+\frac {6}{13} a b^5 x^{13}+\frac {b^6 x^{15}}{15} \]

[Out]

1/3*a^6*x^3+6/5*a^5*b*x^5+15/7*a^4*b^2*x^7+20/9*a^3*b^3*x^9+15/11*a^2*b^4*x^11+6/13*a*b^5*x^13+1/15*b^6*x^15

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {28, 276} \[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {a^6 x^3}{3}+\frac {6}{5} a^5 b x^5+\frac {15}{7} a^4 b^2 x^7+\frac {20}{9} a^3 b^3 x^9+\frac {15}{11} a^2 b^4 x^{11}+\frac {6}{13} a b^5 x^{13}+\frac {b^6 x^{15}}{15} \]

[In]

Int[x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(a^6*x^3)/3 + (6*a^5*b*x^5)/5 + (15*a^4*b^2*x^7)/7 + (20*a^3*b^3*x^9)/9 + (15*a^2*b^4*x^11)/11 + (6*a*b^5*x^13
)/13 + (b^6*x^15)/15

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\int x^2 \left (a b+b^2 x^2\right )^6 \, dx}{b^6} \\ & = \frac {\int \left (a^6 b^6 x^2+6 a^5 b^7 x^4+15 a^4 b^8 x^6+20 a^3 b^9 x^8+15 a^2 b^{10} x^{10}+6 a b^{11} x^{12}+b^{12} x^{14}\right ) \, dx}{b^6} \\ & = \frac {a^6 x^3}{3}+\frac {6}{5} a^5 b x^5+\frac {15}{7} a^4 b^2 x^7+\frac {20}{9} a^3 b^3 x^9+\frac {15}{11} a^2 b^4 x^{11}+\frac {6}{13} a b^5 x^{13}+\frac {b^6 x^{15}}{15} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {a^6 x^3}{3}+\frac {6}{5} a^5 b x^5+\frac {15}{7} a^4 b^2 x^7+\frac {20}{9} a^3 b^3 x^9+\frac {15}{11} a^2 b^4 x^{11}+\frac {6}{13} a b^5 x^{13}+\frac {b^6 x^{15}}{15} \]

[In]

Integrate[x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(a^6*x^3)/3 + (6*a^5*b*x^5)/5 + (15*a^4*b^2*x^7)/7 + (20*a^3*b^3*x^9)/9 + (15*a^2*b^4*x^11)/11 + (6*a*b^5*x^13
)/13 + (b^6*x^15)/15

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.84

method	result	size
default	\(\frac {1}{3} a^{6} x^{3}+\frac {6}{5} a^{5} b \,x^{5}+\frac {15}{7} a^{4} b^{2} x^{7}+\frac {20}{9} a^{3} b^{3} x^{9}+\frac {15}{11} a^{2} b^{4} x^{11}+\frac {6}{13} a \,b^{5} x^{13}+\frac {1}{15} b^{6} x^{15}\)	\(69\)
norman	\(\frac {1}{3} a^{6} x^{3}+\frac {6}{5} a^{5} b \,x^{5}+\frac {15}{7} a^{4} b^{2} x^{7}+\frac {20}{9} a^{3} b^{3} x^{9}+\frac {15}{11} a^{2} b^{4} x^{11}+\frac {6}{13} a \,b^{5} x^{13}+\frac {1}{15} b^{6} x^{15}\)	\(69\)
risch	\(\frac {1}{3} a^{6} x^{3}+\frac {6}{5} a^{5} b \,x^{5}+\frac {15}{7} a^{4} b^{2} x^{7}+\frac {20}{9} a^{3} b^{3} x^{9}+\frac {15}{11} a^{2} b^{4} x^{11}+\frac {6}{13} a \,b^{5} x^{13}+\frac {1}{15} b^{6} x^{15}\)	\(69\)
parallelrisch	\(\frac {1}{3} a^{6} x^{3}+\frac {6}{5} a^{5} b \,x^{5}+\frac {15}{7} a^{4} b^{2} x^{7}+\frac {20}{9} a^{3} b^{3} x^{9}+\frac {15}{11} a^{2} b^{4} x^{11}+\frac {6}{13} a \,b^{5} x^{13}+\frac {1}{15} b^{6} x^{15}\)	\(69\)
gosper	\(\frac {x^{3} \left (3003 b^{6} x^{12}+20790 a \,b^{5} x^{10}+61425 a^{2} b^{4} x^{8}+100100 a^{3} b^{3} x^{6}+96525 a^{4} b^{2} x^{4}+54054 a^{5} b \,x^{2}+15015 a^{6}\right )}{45045}\)	\(71\)

[In]

int(x^2*(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/3*a^6*x^3+6/5*a^5*b*x^5+15/7*a^4*b^2*x^7+20/9*a^3*b^3*x^9+15/11*a^2*b^4*x^11+6/13*a*b^5*x^13+1/15*b^6*x^15

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.83 \[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {1}{15} \, b^{6} x^{15} + \frac {6}{13} \, a b^{5} x^{13} + \frac {15}{11} \, a^{2} b^{4} x^{11} + \frac {20}{9} \, a^{3} b^{3} x^{9} + \frac {15}{7} \, a^{4} b^{2} x^{7} + \frac {6}{5} \, a^{5} b x^{5} + \frac {1}{3} \, a^{6} x^{3} \]

[In]

integrate(x^2*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

1/15*b^6*x^15 + 6/13*a*b^5*x^13 + 15/11*a^2*b^4*x^11 + 20/9*a^3*b^3*x^9 + 15/7*a^4*b^2*x^7 + 6/5*a^5*b*x^5 + 1
/3*a^6*x^3

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.98 \[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {a^{6} x^{3}}{3} + \frac {6 a^{5} b x^{5}}{5} + \frac {15 a^{4} b^{2} x^{7}}{7} + \frac {20 a^{3} b^{3} x^{9}}{9} + \frac {15 a^{2} b^{4} x^{11}}{11} + \frac {6 a b^{5} x^{13}}{13} + \frac {b^{6} x^{15}}{15} \]

[In]

integrate(x**2*(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

a**6*x**3/3 + 6*a**5*b*x**5/5 + 15*a**4*b**2*x**7/7 + 20*a**3*b**3*x**9/9 + 15*a**2*b**4*x**11/11 + 6*a*b**5*x
**13/13 + b**6*x**15/15

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.83 \[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {1}{15} \, b^{6} x^{15} + \frac {6}{13} \, a b^{5} x^{13} + \frac {15}{11} \, a^{2} b^{4} x^{11} + \frac {20}{9} \, a^{3} b^{3} x^{9} + \frac {15}{7} \, a^{4} b^{2} x^{7} + \frac {6}{5} \, a^{5} b x^{5} + \frac {1}{3} \, a^{6} x^{3} \]

[In]

integrate(x^2*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

1/15*b^6*x^15 + 6/13*a*b^5*x^13 + 15/11*a^2*b^4*x^11 + 20/9*a^3*b^3*x^9 + 15/7*a^4*b^2*x^7 + 6/5*a^5*b*x^5 + 1
/3*a^6*x^3

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.83 \[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {1}{15} \, b^{6} x^{15} + \frac {6}{13} \, a b^{5} x^{13} + \frac {15}{11} \, a^{2} b^{4} x^{11} + \frac {20}{9} \, a^{3} b^{3} x^{9} + \frac {15}{7} \, a^{4} b^{2} x^{7} + \frac {6}{5} \, a^{5} b x^{5} + \frac {1}{3} \, a^{6} x^{3} \]

[In]

integrate(x^2*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

1/15*b^6*x^15 + 6/13*a*b^5*x^13 + 15/11*a^2*b^4*x^11 + 20/9*a^3*b^3*x^9 + 15/7*a^4*b^2*x^7 + 6/5*a^5*b*x^5 + 1
/3*a^6*x^3

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.83 \[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {a^6\,x^3}{3}+\frac {6\,a^5\,b\,x^5}{5}+\frac {15\,a^4\,b^2\,x^7}{7}+\frac {20\,a^3\,b^3\,x^9}{9}+\frac {15\,a^2\,b^4\,x^{11}}{11}+\frac {6\,a\,b^5\,x^{13}}{13}+\frac {b^6\,x^{15}}{15} \]

[In]

int(x^2*(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)

[Out]

(a^6*x^3)/3 + (b^6*x^15)/15 + (6*a^5*b*x^5)/5 + (6*a*b^5*x^13)/13 + (15*a^4*b^2*x^7)/7 + (20*a^3*b^3*x^9)/9 +
(15*a^2*b^4*x^11)/11